0.8 Loans: The Conventional Mortgage
Mortgages are different than ordinary loans. In most loans, interest is paid over the term or life of the loan, and the entire principal is paid in one fell swoop at the loan’s term or maturity. Most mortgages are self-amortizing, which means that all payments include portions of both interest and principal, resulting in decreasing principal balances over time until, at maturity, the entire loan will have been paid off. Let us see, by way of example, how this may work.
| Given: | Principal: | $100,000 | Rate: | 9% |
| Term: | 10 years | Period: | Yearly |
Mathematical Rationale:
The loan proceeds, i.e., $100,000 in this case, represent the present value. The “periodic payment” represents the annuity payments to be made over future years. The present value of the annuity payments should equal the loan principal:
Principal = Periodic Payment x Present Value Annuity Factor
Calculation:
Periodic Payment = Principal/PV Annuity Factor
$100,000 ÷ 6.42 = $ 15,576.32
Interest = Opening Balance x Rate
Principal Payment = Periodic (“Total Cash”) Payment less Interest
Balance = Opening Balance less Principal Payment
Payment and Amortization Schedule:
| Cash Payments | ||||
| Year | Total Payments | Interest | Amortization | Balance |
| 0 | $100,000 | |||
| 1 | $15,576 | $9,000 | $6,576 | $93,424 |
| 2 | 15,576 | 8,408 | 7,168 | 86,256 |
| 3 | 15,576 | 7,763 | 7,813 | 78,443 |
| 4 | 15,576 | 7,060 | 8,516 | 69,927 |
| 5 | 15,576 | 6,293 | 9,283 | 60,644 |
| 6 | 15,576 | 5,458 | 10,118 | 50,526 |
| 7 | 15,576 | 4,547 | 11,029 | 39,497 |
| 8 | 15,576 | 3,555 | 12,021 | 27,476 |
| 9 | 15,576 | 2,473 | 13,103 | 14,373 |
| 10 | 15,576 | 1,294 | 14,282 | 0 |
| Totals | $155,763 | $55,763 | $100,000 |
The 9% interest payments are charged against the balance of the loan. In the first year, the interest portion of the payment is 9% of $100,000 or $9,000. This leaves $15,576 less $9,000 = $6,576 going toward principal reduction, i.e., amortization. The new, amortized balance is hence $100,000 less $6,576 = $93,424. This continues for each period until maturity.
While you may be thinking in terms of borrowing, investors may invest their money in mortgage-backed securities, e.g., bonds issued by the Government National Mortgage Association (or Ginnie Mae). These investors are, in effect, lenders, the other side, as it were, of the same coin. The math is the same.
Note:
Mortgage payments are usually made in MONTHLY installments, and often with greater maturities. This has been simplified for illustration purposes, so that the reader may easily refer to standard interest rate tables.
The mortgage formula is important to master as it will be used again in three additional contexts: 1. Leasing; 2. Bond Accounting; and 3. Capital Budgeting: The Annual Annuity Approach.